In recent years, there have been proposed and developed various oil pump control system equipped automotive vehicles in which a sub oil pump (e.g., an electric-motor-driven oil pump) is utilized to compensate for a lack in discharge pressure of a main oil pump (e.g., a mechanical oil pump driven by a driving power source). For instance, when a pressure level of hydraulic pressure needed for friction engagement elements of an automatic transmission reduces to below a given pressure value or when a revolution speed of the driving power source reduces to below a given speed value, the oil pump control system is configured to start up the sub oil pump (the electric-motor-driven oil pump). One such oil pump control system has been disclosed in Japanese Patent Provisional Publication No. 2002-206634 (hereinafter is referred to as “JP2002-206634”), corresponding to U.S. Pat. No. 6,769,502, issued on Aug. 3, 2004.